Analysis: learning session
Gamified Feedback in Adaptive Retrieval Practice
Maarten van der Velde & Gesa van den Broek
Last updated: 2024-11-20
Setup
library(here)here() starts at /Users/maarten/Documents/projects/gamified-feedback-studylibrary(dplyr)
Attaching package: 'dplyr'The following objects are masked from 'package:stats':
filter, lagThe following objects are masked from 'package:base':
intersect, setdiff, setequal, unionlibrary(ggplot2)
library(scales)
library(patchwork)Warning: package 'patchwork' was built under R version 4.3.3library(stringr)
library(tidyr)
library(lme4)Loading required package: Matrix
Attaching package: 'Matrix'The following objects are masked from 'package:tidyr':
expand, pack, unpacklibrary(lmerTest)
Attaching package: 'lmerTest'The following object is masked from 'package:lme4':
lmerThe following object is masked from 'package:stats':
steplibrary(brms)Loading required package: RcppLoading 'brms' package (version 2.21.0). Useful instructions
can be found by typing help('brms'). A more detailed introduction
to the package is available through vignette('brms_overview').
Attaching package: 'brms'The following object is masked from 'package:lme4':
ngrpsThe following object is masked from 'package:stats':
ar# Set up parallel processing for Bayesian models
library(future)
plan(multisession, workers = 4)Helper functions for plots and tables:
source(here("scripts", "00_visualisation_functions.R"))Loading required package: data.table
Attaching package: 'data.table'The following objects are masked from 'package:dplyr':
between, first, lastLoading required package: flextableLoad processed data:
d_learn <- readRDS(here("data", "processed", "d_learn.rds"))add_experiment_cols <- function (data) {
data |>
mutate(exp_order = case_when(
gamified_first == 0 & exp_group == "score" ~ "Control—Score",
gamified_first == 0 & exp_group == "both" ~ "Control—Both",
gamified_first == 1 & exp_group == "score" ~ "Score—Control",
gamified_first == 1 & exp_group == "both" ~ "Both—Control"
)) |>
mutate(type = ifelse(gamified, "Gamified", "Control"))
}Helper function for interpreting Bayes factors:
bf_to_strength <- function (bf) {
direction <- "for"
if (bf < 1) {
bf <- 1/bf
direction <- "against"
}
strength <- case_when(
bf == 1 ~ "No",
bf < 3 ~ "Anecdotal",
bf < 10 ~ "Moderate",
bf < 30 ~ "Strong",
bf < 100 ~ "Very strong",
TRUE ~ "Extreme"
)
paste0(strength, " evidence ", direction)
}Does gamification change performance during practice?
Accuracy
Prepare data
d_learn_acc <- d_learn |>
filter(!study_trial) |>
group_by(subject, exp_group, block, condition, gamified, gamified_first) |>
summarise(accuracy = mean(correct))`summarise()` has grouped output by 'subject', 'exp_group', 'block',
'condition', 'gamified'. You can override using the `.groups` argument.d_learn_acc_agg <- d_learn_acc |>
group_by(block, condition, gamified, gamified_first, exp_group) |>
summarise(acc = mean(accuracy, na.rm = T),
acc_se = sd(accuracy, na.rm = T)/sqrt(n())) |>
ungroup() |>
add_experiment_cols()`summarise()` has grouped output by 'block', 'condition', 'gamified',
'gamified_first'. You can override using the `.groups` argument.Visualise data
p_learn_acc <- plot_data(d_learn_acc_agg, acc, acc_se, "Accuracy") +
scale_y_continuous(limits = c(.725, .875), labels = scales::percent_format())
p_learn_acc
Fit frequentist model
Prepare data for modelling by mean-centering categorical predictors:
d_learn_m <- d_learn |>
filter(!study_trial) |>
mutate(exp_group_c = ifelse(exp_group == "score", 0, 1),
exp_group_c = exp_group_c - mean(exp_group_c),
gamified_first_c = gamified_first - mean(gamified_first))m_learn_acc <- glmer(correct ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject) + (1 | fact),
family = "binomial",
data = d_learn_m)
summary(m_learn_acc)Generalized linear mixed model fit by maximum likelihood (Laplace
Approximation) [glmerMod]
Family: binomial ( logit )
Formula:
correct ~ gamified + gamified:exp_group_c + gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c + (1 | subject) + (1 | fact)
Data: d_learn_m
AIC BIC logLik deviance df.resid
38686.5 38772.8 -19333.3 38666.5 41479
Scaled residuals:
Min 1Q Median 3Q Max
-5.4230 0.2697 0.4166 0.5308 0.8948
Random effects:
Groups Name Variance Std.Dev.
subject (Intercept) 0.2932 0.5415
fact (Intercept) 0.1178 0.3432
Number of obs: 41489, groups: subject, 166; fact, 78
Fixed effects:
Estimate Std. Error z value
(Intercept) 1.621184 0.060640 26.735
gamifiedTRUE -0.007472 0.025932 -0.288
gamifiedFALSE:exp_group_c -0.181824 0.092172 -1.973
gamifiedTRUE:exp_group_c -0.160723 0.092242 -1.742
gamifiedFALSE:gamified_first_c 0.069727 0.092386 0.755
gamifiedTRUE:gamified_first_c -0.079241 0.092419 -0.857
gamifiedFALSE:exp_group_c:gamified_first_c -0.245058 0.185121 -1.324
gamifiedTRUE:exp_group_c:gamified_first_c -0.181772 0.185123 -0.982
Pr(>|z|)
(Intercept) <2e-16 ***
gamifiedTRUE 0.7732
gamifiedFALSE:exp_group_c 0.0485 *
gamifiedTRUE:exp_group_c 0.0814 .
gamifiedFALSE:gamified_first_c 0.4504
gamifiedTRUE:gamified_first_c 0.3912
gamifiedFALSE:exp_group_c:gamified_first_c 0.1856
gamifiedTRUE:exp_group_c:gamified_first_c 0.3261
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) gmTRUE gmfdFALSE:x__ gmfdTRUE:x__ gmfdFALSE:g__
gamifidTRUE -0.215
gmfdFALSE:x__ -0.014 0.017
gmfdTRUE:x__ -0.008 -0.012 0.841
gmfdFALSE:g__ -0.011 0.000 -0.010 -0.004
gmfdTRUE:g__ -0.014 0.001 -0.004 -0.009 0.842
gFALSE:__:_ -0.008 0.011 -0.017 -0.016 -0.019
gTRUE:__:__ -0.003 -0.009 -0.016 -0.018 -0.010
gmfdTRUE:g__ gFALSE:__:
gamifidTRUE
gmfdFALSE:x__
gmfdTRUE:x__
gmfdFALSE:g__
gmfdTRUE:g__
gFALSE:__:_ -0.010
gTRUE:__:__ -0.016 0.842 print_model_table(m_learn_acc)Effect | Estimate | SE | z-value | p-value |
|---|---|---|---|---|
Intercept (Control) | 1.62 | 0.06 | 26.73 | <.001 |
Gamification | -0.01 | 0.03 | -0.29 | .773 |
Type of gamification (Points-and-progress - Points) | -0.16 | 0.09 | -1.74 | .081 |
Type of gamification on Control (Points-and-progress - Points) | -0.18 | 0.09 | -1.97 | .049 |
Block on gamification (1 - 2) | -0.08 | 0.09 | -0.86 | .391 |
Block on Control (2 - 1) | 0.07 | 0.09 | 0.75 | .450 |
Type of gamification; block difference (Block 1 - Block 2) | -0.18 | 0.19 | -0.98 | .326 |
Type of gamification on Control; block difference (Block 2 - Block 1) | -0.25 | 0.19 | -1.32 | .186 |
Fitted values
d_model_fit <- crossing(
gamified = FALSE,
exp_group_c = sort(unique(d_learn_m$exp_group_c)),
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_learn_acc,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitVisualise fitted model
p_learn_acc_m <- plot_model_fit(m_learn_acc, d_learn_m, y_lab = "Accuracy") +
scale_y_continuous(limits = c(.75, .90), labels = scales::percent_format(accuracy = .1)) block condition gamified gamified_first exp_group gamified_first_c
1 1 Control FALSE FALSE both -0.5320687
2 1 Control FALSE FALSE score -0.5320687
3 1 Points TRUE TRUE score 0.4679313
4 1 Points-and-progress TRUE TRUE both 0.4679313
5 2 Control FALSE TRUE both 0.4679313
6 2 Control FALSE TRUE score 0.4679313
7 2 Points TRUE FALSE score -0.5320687
8 2 Points-and-progress TRUE FALSE both -0.5320687
exp_group_c pred_val exp_order type
1 0.5296826 0.8258994 Control—Both Control
2 -0.4703174 0.8331719 Control—Score Control
3 -0.4703174 0.8445168 Score—Control Gamified
4 0.5296826 0.8094510 Both—Control Gamified
5 0.5296826 0.8170913 Both—Control Control
6 -0.4703174 0.8573262 Score—Control Control
7 -0.4703174 0.8436944 Control—Score Gamified
8 0.5296826 0.8350666 Control—Both GamifiedScale for y is already present.
Adding another scale for y, which will replace the existing scale.p_learn_acc_m
Fit Bayesian model
Fit again with brms so that we can calculate Bayes
Factors. Because we expect any fixed effects to be at most moderate in
size, we will use a weakly informative Normal(0, 1) prior for these
effects.
m_learn_acc_brms <- brm(correct ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject) + (1 | fact),
family = "bernoulli",
data = d_learn_m,
prior = set_prior("normal(0, 1)", class = "b"),
chains = 4,
iter = 11000,
warmup = 1000,
sample_prior = TRUE,
future = TRUE,
seed = 0)Compiling Stan program...Trying to compile a simple C fileRunning /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make[1]: *** [foo.o] Error 1Start sampling
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Chain 4: summary(m_learn_acc_brms) Family: bernoulli
Links: mu = logit
Formula: correct ~ gamified + gamified:exp_group_c + gamified:gamified_first_c + gamified:gamified_first_c:exp_group_c + (1 | subject) + (1 | fact)
Data: d_learn_m (Number of observations: 41489)
Draws: 4 chains, each with iter = 11000; warmup = 1000; thin = 1;
total post-warmup draws = 40000
Multilevel Hyperparameters:
~fact (Number of levels: 78)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 0.35 0.04 0.29 0.43 1.00 8273 14730
~subject (Number of levels: 166)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 0.55 0.04 0.49 0.63 1.00 7175 11801
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI
Intercept 1.62 0.06 1.50 1.75
gamifiedTRUE -0.01 0.03 -0.06 0.04
gamifiedFALSE:exp_group_c -0.18 0.09 -0.36 0.00
gamifiedTRUE:exp_group_c -0.16 0.09 -0.34 0.02
gamifiedFALSE:gamified_first_c 0.07 0.09 -0.11 0.25
gamifiedTRUE:gamified_first_c -0.08 0.09 -0.26 0.11
gamifiedFALSE:exp_group_c:gamified_first_c -0.23 0.18 -0.60 0.12
gamifiedTRUE:exp_group_c:gamified_first_c -0.17 0.18 -0.53 0.19
Rhat Bulk_ESS Tail_ESS
Intercept 1.00 5005 10011
gamifiedTRUE 1.00 43753 29667
gamifiedFALSE:exp_group_c 1.00 5185 9292
gamifiedTRUE:exp_group_c 1.00 5241 9538
gamifiedFALSE:gamified_first_c 1.00 5024 10205
gamifiedTRUE:gamified_first_c 1.00 4946 9580
gamifiedFALSE:exp_group_c:gamified_first_c 1.00 5514 10973
gamifiedTRUE:exp_group_c:gamified_first_c 1.00 5556 10691
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Inspect the posterior sample distributions of the fixed effects:
plot(m_learn_acc_brms, nvariables = 8, variable = "^b_", regex = TRUE)
Bayes factors
Do a hypothesis test for all fixed-effect coefficients (both main
effects and interactions) in the model being equal to zero. The column
Evid.Ratio shows the Bayes Factor in favour of the null
hypothesis (\(BF_{01}\)).
h_learn_acc <- hypothesis(m_learn_acc_brms,
c("gamifiedTRUE = 0",
"gamifiedFALSE:exp_group_c = 0",
"gamifiedTRUE:exp_group_c = 0",
"gamifiedFALSE:gamified_first_c = 0",
"gamifiedTRUE:gamified_first_c = 0",
"gamifiedFALSE:exp_group_c:gamified_first_c = 0",
"gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
class = "b")
h_learn_acc$hypothesis |>
mutate(BF10 = 1 / Evid.Ratio,
evidence_for_null = sapply(Evid.Ratio, bf_to_strength))This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line). Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.
sd_ratio_acc <- plot(h_learn_acc, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)
sd_ratio_acc[[1]] +
geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
Conclusion
The Bayesian model finds anecdotal to strong evidence in favour of the null hypothesis (no effect on learning accuracy) for each of the regression coefficients.
Response time
Response time on correct answers only.
Prepare data
To keep the visualisation of average response times by condition simple, we calculate the median RT per participant, and then take the mean and SD of these medians (which are themselves roughly normally distributed).
d_learn_rt <- d_learn |>
filter(!study_trial) |>
filter(correct) |>
mutate(rt = rt / 1000) |>
group_by(subject, exp_group, block, condition, gamified, gamified_first) |>
summarise(rt_median = median(rt, na.rm = TRUE))`summarise()` has grouped output by 'subject', 'exp_group', 'block',
'condition', 'gamified'. You can override using the `.groups` argument.d_learn_rt_agg <- d_learn_rt |>
group_by(block, condition, gamified, gamified_first, exp_group) |>
summarise(rt_mean = mean(rt_median, na.rm = T),
rt_se = sd(rt_median, na.rm = T)/sqrt(n())) |>
ungroup() |>
add_experiment_cols()`summarise()` has grouped output by 'block', 'condition', 'gamified',
'gamified_first'. You can override using the `.groups` argument.Visualise data
p_learn_rt <- plot_data(d_learn_rt_agg, rt_mean, rt_se, "Response time (s)") +
scale_y_continuous(limits = c(1.3, 1.8), labels = scales::comma_format())
p_learn_rt
Fit frequentist model
Since RT data is not normally distributed, we fit a lognormal model to the response times. (See https://bbolker.github.io/mixedmodels-misc/glmmFAQ.html#gamma-glmms .) Prepare data for modelling by mean-centering categorical predictors:
d_learn_rt_m <- d_learn |>
filter(!study_trial) |>
filter(correct) |>
mutate(log_rt = log(rt / 1000)) |>
mutate(exp_group_c = ifelse(exp_group == "score", 0, 1),
exp_group_c = exp_group_c - mean(exp_group_c),
gamified_first_c = gamified_first - mean(gamified_first)
)m_learn_rt <- lmer(log_rt ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject) + (1 | fact),
data = d_learn_rt_m)
summary(m_learn_rt)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
log_rt ~ gamified + gamified:exp_group_c + gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c + (1 | subject) + (1 | fact)
Data: d_learn_rt_m
REML criterion at convergence: 58081.7
Scaled residuals:
Min 1Q Median 3Q Max
-15.6337 -0.5696 -0.1470 0.4864 4.8537
Random effects:
Groups Name Variance Std.Dev.
subject (Intercept) 0.03589 0.1895
fact (Intercept) 0.01190 0.1091
Residual 0.32134 0.5669
Number of obs: 33657, groups: subject, 166; fact, 78
Fixed effects:
Estimate Std. Error df
(Intercept) 4.837e-01 1.973e-02 2.287e+02
gamifiedTRUE 1.734e-02 6.240e-03 3.347e+04
gamifiedFALSE:exp_group_c 9.712e-03 3.077e-02 1.754e+02
gamifiedTRUE:exp_group_c 4.371e-02 3.078e-02 1.757e+02
gamifiedFALSE:gamified_first_c 1.335e-03 3.084e-02 1.753e+02
gamifiedTRUE:gamified_first_c 7.474e-02 3.085e-02 1.754e+02
gamifiedFALSE:exp_group_c:gamified_first_c -3.774e-02 6.175e-02 1.755e+02
gamifiedTRUE:exp_group_c:gamified_first_c -2.206e-02 6.175e-02 1.754e+02
t value Pr(>|t|)
(Intercept) 24.522 < 2e-16 ***
gamifiedTRUE 2.779 0.00546 **
gamifiedFALSE:exp_group_c 0.316 0.75264
gamifiedTRUE:exp_group_c 1.420 0.15738
gamifiedFALSE:gamified_first_c 0.043 0.96553
gamifiedTRUE:gamified_first_c 2.423 0.01640 *
gamifiedFALSE:exp_group_c:gamified_first_c -0.611 0.54185
gamifiedTRUE:exp_group_c:gamified_first_c -0.357 0.72138
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) gmTRUE gmfdFALSE:x__ gmfdTRUE:x__ gmfdFALSE:g__
gamifidTRUE -0.158
gmfdFALSE:x__ -0.020 0.000
gmfdTRUE:x__ -0.020 0.000 0.917
gmfdFALSE:g__ -0.017 0.005 -0.003 -0.004
gmfdTRUE:g__ -0.016 0.006 -0.004 -0.003 0.918
gFALSE:__:_ -0.002 -0.002 -0.022 -0.021 -0.025
gTRUE:__:__ -0.003 0.002 -0.021 -0.018 -0.025
gmfdTRUE:g__ gFALSE:__:
gamifidTRUE
gmfdFALSE:x__
gmfdTRUE:x__
gmfdFALSE:g__
gmfdTRUE:g__
gFALSE:__:_ -0.025
gTRUE:__:__ -0.025 0.917 print_model_table(m_learn_rt)Effect | Estimate | SE | df | t-value | p-value |
|---|---|---|---|---|---|
Intercept (Control) | 0.484 | 0.020 | 228.75 | 24.52 | <.001 |
Gamification | 0.017 | 0.006 | 33,473.10 | 2.78 | .005 |
Type of gamification (Points-and-progress - Points) | 0.044 | 0.031 | 175.68 | 1.42 | .157 |
Type of gamification on Control (Points-and-progress - Points) | 0.010 | 0.031 | 175.40 | 0.32 | .753 |
Block on gamification (1 - 2) | 0.075 | 0.031 | 175.37 | 2.42 | .016 |
Block on Control (2 - 1) | 0.001 | 0.031 | 175.29 | 0.04 | .966 |
Type of gamification; block difference (Block 1 - Block 2) | -0.022 | 0.062 | 175.44 | -0.36 | .721 |
Type of gamification on Control; block difference (Block 2 - Block 1) | -0.038 | 0.062 | 175.48 | -0.61 | .542 |
Fitted values
d_model_fit <- crossing(
gamified = c(FALSE, TRUE),
exp_group_c = 0,
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_learn_rt,
newdata = d_model_fit,
re.form = NA,
type = "response") |>
exp() # Transform logRT to RT
d_model_fitd_model_fit <- crossing(
gamified = c(FALSE, TRUE),
exp_group_c = 0,
gamified_first_c = sort(unique(d_learn_rt_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_learn_rt,
newdata = d_model_fit,
re.form = NA,
type = "response") |>
exp() # Transform logRT to RT
d_model_fitVisualise fitted model
p_learn_rt_m <- plot_model_fit(m_learn_rt, d_learn_rt_m, exp_trans = TRUE, y_lab = "Response time (s)") +
scale_y_continuous(limits = c(1.4, 1.9), labels = scales::comma_format()) block condition gamified gamified_first exp_group gamified_first_c
1 1 Control FALSE FALSE both -0.5311822
2 1 Control FALSE FALSE score -0.5311822
3 1 Points TRUE TRUE score 0.4688178
4 1 Points-and-progress TRUE TRUE both 0.4688178
5 2 Control FALSE TRUE both 0.4688178
6 2 Control FALSE TRUE score 0.4688178
7 2 Points TRUE FALSE score -0.5311822
8 2 Points-and-progress TRUE FALSE both -0.5311822
exp_group_c pred_val exp_order type
1 0.5378079 1.647124 Control—Both Control
2 -0.4621921 1.598827 Control—Score Control
3 -0.4621921 1.683192 Score—Control Gamified
4 0.5378079 1.740307 Both—Control Gamified
5 0.5378079 1.616182 Both—Control Control
6 -0.4621921 1.629135 Score—Control Control
7 -0.4621921 1.546128 Control—Score Gamified
8 0.5378079 1.634243 Control—Both GamifiedScale for y is already present.
Adding another scale for y, which will replace the existing scale.p_learn_rt_m
Fit Bayesian model
Fit again with brms so that we can calculate Bayes
Factors. Because we expect any fixed effects to be at most moderate in
size, we will use a weakly informative Normal(0, .1) prior for these
effects.
m_learn_rt_brms <- brm(log_rt ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject) + (1 | fact),
family = "gaussian",
data = d_learn_rt_m,
prior = set_prior("normal(0, .1)", class = "b"),
chains = 4,
iter = 11000,
warmup = 1000,
sample_prior = TRUE,
future = TRUE,
seed = 0)Compiling Stan program...Trying to compile a simple C fileRunning /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make[1]: *** [foo.o] Error 1Start sampling
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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Chain 4: summary(m_learn_rt_brms) Family: gaussian
Links: mu = identity; sigma = identity
Formula: log_rt ~ gamified + gamified:exp_group_c + gamified:gamified_first_c + gamified:gamified_first_c:exp_group_c + (1 | subject) + (1 | fact)
Data: d_learn_rt_m (Number of observations: 33657)
Draws: 4 chains, each with iter = 11000; warmup = 1000; thin = 1;
total post-warmup draws = 40000
Multilevel Hyperparameters:
~fact (Number of levels: 78)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 0.11 0.01 0.09 0.13 1.00 8718 16292
~subject (Number of levels: 166)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 0.19 0.01 0.17 0.21 1.00 7137 14437
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI
Intercept 0.48 0.02 0.44 0.52
gamifiedTRUE 0.02 0.01 0.01 0.03
gamifiedFALSE:exp_group_c 0.01 0.03 -0.05 0.06
gamifiedTRUE:exp_group_c 0.04 0.03 -0.02 0.10
gamifiedFALSE:gamified_first_c -0.01 0.03 -0.06 0.05
gamifiedTRUE:gamified_first_c 0.07 0.03 0.01 0.12
gamifiedFALSE:exp_group_c:gamified_first_c -0.02 0.05 -0.12 0.07
gamifiedTRUE:exp_group_c:gamified_first_c -0.01 0.05 -0.10 0.09
Rhat Bulk_ESS Tail_ESS
Intercept 1.00 4838 9834
gamifiedTRUE 1.00 101540 27412
gamifiedFALSE:exp_group_c 1.00 5027 10328
gamifiedTRUE:exp_group_c 1.00 4972 10050
gamifiedFALSE:gamified_first_c 1.00 4675 9141
gamifiedTRUE:gamified_first_c 1.00 4765 9731
gamifiedFALSE:exp_group_c:gamified_first_c 1.00 6758 11635
gamifiedTRUE:exp_group_c:gamified_first_c 1.00 6701 12077
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 0.57 0.00 0.56 0.57 1.00 88321 27944
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Inspect the posterior sample distributions of the fixed effects:
plot(m_learn_rt_brms, nvariables = 8, variable = "^b_", regex = TRUE)
Bayes factors
Do a hypothesis learn for all fixed-effect coefficients (both main
effects and interactions) in the model being equal to zero. The column
Evid.Ratio shows the Bayes Factor in favour of the null
hypothesis (\(BF_{01}\)).
h_learn_rt <- hypothesis(m_learn_rt_brms,
c("gamifiedTRUE = 0",
"gamifiedFALSE:exp_group_c = 0",
"gamifiedTRUE:exp_group_c = 0",
"gamifiedFALSE:gamified_first_c = 0",
"gamifiedTRUE:gamified_first_c = 0",
"gamifiedFALSE:exp_group_c:gamified_first_c = 0",
"gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
class = "b")
h_learn_rt$hypothesis |>
mutate(BF10 = 1 / Evid.Ratio,
evidence_for_null = sapply(Evid.Ratio, bf_to_strength))This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line). Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.
sd_ratio_rt <- plot(h_learn_rt, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)
sd_ratio_rt[[1]] +
geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
Fitted values
d_model_fit <- crossing(
gamified = c(FALSE, TRUE),
exp_group_c = 0,
gamified_first_c = sort(unique(d_learn_rt_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_learn_rt_brms,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1] |>
exp() # Transform logRT to RT
d_model_fitConclusion
The Bayesian model finds anecdotal to strong evidence in favour of the null hypothesis (no effect on correct RT) for each of the regression coefficients.
Total score
The total score is the number of points after the last trial in a block.
Prepare data
d_learn_score <- d_learn |>
group_by(subject, exp_group, block, condition, gamified, gamified_first) |>
slice(n())
d_learn_score_agg <- d_learn_score |>
group_by(block, condition, gamified, gamified_first, exp_group) |>
summarise(feedback_score_mean = mean(feedback_score, na.rm = T),
feedback_score_se = sd(feedback_score, na.rm = T)/sqrt(n())) |>
ungroup() |>
add_experiment_cols()`summarise()` has grouped output by 'block', 'condition', 'gamified',
'gamified_first'. You can override using the `.groups` argument.Visualise data
p_learn_score <- plot_data(d_learn_score_agg, feedback_score_mean, feedback_score_se, "Total score") +
scale_y_continuous(limits = c(1000, 1400), labels = scales::comma_format())
p_learn_score
Distribution of scores:
p_learn_score_dist <- ggplot(d_learn_score, aes(x = feedback_score, fill = condition)) +
facet_grid(condition ~ .) +
geom_histogram(aes(y=..density..), colour = "black", binwidth = 100) +
geom_density(alpha = .5) +
geom_vline(xintercept = c(1200, 1500), lty = 2) +
scale_fill_manual(values = col_condition) +
scale_colour_manual(values = col_condition) +
guides(fill = "none",
colour = "none") +
labs(x = "Total score",
y = "Density") +
theme_paper
p_learn_score_distWarning: The dot-dot notation (`..density..`) was deprecated in ggplot2 3.4.0.
ℹ Please use `after_stat(density)` instead.
This warning is displayed once every 8 hours.
Call `lifecycle::last_lifecycle_warnings()` to see where this warning was generated.
ggsave(here("output", "practice_scores.png"), width = 7.5, height = 5)Fit frequentist model
Prepare data for modelling by mean-centering categorical predictors:
d_learn_score_m <- d_learn_score |>
ungroup() |>
mutate(exp_group_c = ifelse(exp_group == "score", 0, 1),
exp_group_c = exp_group_c - mean(exp_group_c),
gamified_first_c = gamified_first - mean(gamified_first))m_learn_score <- lmer(feedback_score ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = d_learn_score_m)
summary(m_learn_score)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
feedback_score ~ gamified + gamified:exp_group_c + gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: d_learn_score_m
REML criterion at convergence: 4534.5
Scaled residuals:
Min 1Q Median 3Q Max
-2.48749 -0.54087 -0.03256 0.49006 2.49651
Random effects:
Groups Name Variance Std.Dev.
subject (Intercept) 67352 259.5
Residual 25509 159.7
Number of obs: 332, groups: subject, 166
Fixed effects:
Estimate Std. Error df t value
(Intercept) 1245.256 23.652 212.312 52.649
gamifiedTRUE -4.856 17.531 162.000 -0.277
gamifiedFALSE:exp_group_c -72.770 47.359 212.312 -1.537
gamifiedTRUE:exp_group_c -72.704 47.359 212.312 -1.535
gamifiedFALSE:gamified_first_c 7.425 47.473 212.312 0.156
gamifiedTRUE:gamified_first_c -123.533 47.473 212.312 -2.602
gamifiedFALSE:exp_group_c:gamified_first_c -74.168 95.059 212.312 -0.780
gamifiedTRUE:exp_group_c:gamified_first_c -85.754 95.059 212.312 -0.902
Pr(>|t|)
(Intercept) < 2e-16 ***
gamifiedTRUE 0.78215
gamifiedFALSE:exp_group_c 0.12589
gamifiedTRUE:exp_group_c 0.12623
gamifiedFALSE:gamified_first_c 0.87586
gamifiedTRUE:gamified_first_c 0.00992 **
gamifiedFALSE:exp_group_c:gamified_first_c 0.43613
gamifiedTRUE:exp_group_c:gamified_first_c 0.36802
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) gmTRUE gmfdFALSE:x__ gmfdTRUE:x__ gmfdFALSE:g__
gamifidTRUE -0.371
gmfdFALSE:x__ 0.000 0.000
gmfdTRUE:x__ 0.000 0.000 0.725
gmfdFALSE:g__ 0.000 0.000 -0.004 -0.003
gmfdTRUE:g__ 0.000 0.000 -0.003 -0.004 0.725
gFALSE:__:_ -0.004 0.002 0.000 0.000 0.001
gTRUE:__:__ -0.003 -0.002 0.000 0.000 0.001
gmfdTRUE:g__ gFALSE:__:
gamifidTRUE
gmfdFALSE:x__
gmfdTRUE:x__
gmfdFALSE:g__
gmfdTRUE:g__
gFALSE:__:_ 0.001
gTRUE:__:__ 0.001 0.725 print_model_table(m_learn_score)Effect | Estimate | SE | df | t-value | p-value |
|---|---|---|---|---|---|
Intercept (Control) | 1,245.256 | 23.652 | 212.31 | 52.65 | <.001 |
Gamification | -4.856 | 17.531 | 162.00 | -0.28 | .782 |
Type of gamification (Points-and-progress - Points) | -72.704 | 47.359 | 212.31 | -1.54 | .126 |
Type of gamification on Control (Points-and-progress - Points) | -72.770 | 47.359 | 212.31 | -1.54 | .126 |
Block on gamification (1 - 2) | -123.533 | 47.473 | 212.31 | -2.60 | .010 |
Block on Control (2 - 1) | 7.425 | 47.473 | 212.31 | 0.16 | .876 |
Type of gamification; block difference (Block 1 - Block 2) | -85.754 | 95.059 | 212.31 | -0.90 | .368 |
Type of gamification on Control; block difference (Block 2 - Block 1) | -74.168 | 95.059 | 212.31 | -0.78 | .436 |
Fitted values
d_model_fit <- crossing(
gamified = TRUE,
exp_group_c = 0,
gamified_first_c = sort(unique(d_learn_score_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_learn_score,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitFit Bayesian model
Since score is on a much larger scale than other dependent variables, we expect coefficients for fixed effects to be on a much larger scale too. Adjust the prior accordingly:
m_learn_score_bayes <- brm(feedback_score ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
family = "gaussian",
data = d_learn_score_m,
prior = set_prior("normal(0, 100)", class = "b"),
chains = 4,
iter = 11000,
warmup = 1000,
sample_prior = TRUE,
future = TRUE,
seed = 0)Compiling Stan program...Trying to compile a simple C fileRunning /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make[1]: *** [foo.o] Error 1Start sampling
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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Chain 4: summary(m_learn_score_bayes) Family: gaussian
Links: mu = identity; sigma = identity
Formula: feedback_score ~ gamified + gamified:exp_group_c + gamified:gamified_first_c + gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: d_learn_score_m (Number of observations: 332)
Draws: 4 chains, each with iter = 11000; warmup = 1000; thin = 1;
total post-warmup draws = 40000
Multilevel Hyperparameters:
~subject (Number of levels: 166)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 259.54 17.49 227.10 295.57 1.00 11131 20038
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI
Intercept 1245.28 23.67 1198.74 1291.38
gamifiedTRUE -4.70 17.40 -38.90 29.50
gamifiedFALSE:exp_group_c -51.79 40.99 -133.20 28.87
gamifiedTRUE:exp_group_c -51.82 41.01 -132.72 29.14
gamifiedFALSE:gamified_first_c 19.51 41.27 -61.15 100.19
gamifiedTRUE:gamified_first_c -103.71 41.42 -184.36 -21.94
gamifiedFALSE:exp_group_c:gamified_first_c -26.03 63.82 -151.67 98.47
gamifiedTRUE:exp_group_c:gamified_first_c -35.52 64.01 -160.77 90.96
Rhat Bulk_ESS Tail_ESS
Intercept 1.00 12041 20700
gamifiedTRUE 1.00 85749 28038
gamifiedFALSE:exp_group_c 1.00 13502 22943
gamifiedTRUE:exp_group_c 1.00 13959 22394
gamifiedFALSE:gamified_first_c 1.00 13528 22841
gamifiedTRUE:gamified_first_c 1.00 13690 22803
gamifiedFALSE:exp_group_c:gamified_first_c 1.00 23052 28888
gamifiedTRUE:exp_group_c:gamified_first_c 1.00 22372 28974
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 160.88 9.02 144.43 179.62 1.00 18231 26019
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Inspect the posterior sample distributions of the fixed effects:
plot(m_learn_score_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
Bayes factors
Do a hypothesis learn for all fixed-effect coefficients (both main
effects and interactions) in the model being equal to zero. The column
Evid.Ratio shows the Bayes Factor in favour of the null
hypothesis (\(BF_{01}\)).
h_learn_score <- hypothesis(m_learn_score_bayes,
c("gamifiedTRUE = 0",
"gamifiedFALSE:exp_group_c = 0",
"gamifiedTRUE:exp_group_c = 0",
"gamifiedFALSE:gamified_first_c = 0",
"gamifiedTRUE:gamified_first_c = 0",
"gamifiedFALSE:exp_group_c:gamified_first_c = 0",
"gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
class = "b")
h_learn_score$hypothesis |>
mutate(BF10 = 1 / Evid.Ratio,
evidence_for_null = sapply(Evid.Ratio, bf_to_strength))This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line). Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.
sd_ratio_score <- plot(h_learn_score, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)
sd_ratio_score[[1]] +
geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
Fitted values
d_model_fit <- crossing(
gamified = TRUE,
exp_group_c = 0,
gamified_first_c = sort(unique(d_learn_score_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_learn_score_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitNumber of words practiced
Prepare data
d_learn_words <- d_learn |>
group_by(subject, exp_group, block, condition, gamified, gamified_first) |>
summarise(words_seen = n_distinct(fact))`summarise()` has grouped output by 'subject', 'exp_group', 'block',
'condition', 'gamified'. You can override using the `.groups` argument.d_learn_words_agg <- d_learn_words |>
group_by(block, condition, gamified, gamified_first, exp_group) |>
summarise(words_mean = mean(words_seen, na.rm = T),
words_se = sd(words_seen, na.rm = T)/sqrt(n())) |>
ungroup() |>
add_experiment_cols()`summarise()` has grouped output by 'block', 'condition', 'gamified',
'gamified_first'. You can override using the `.groups` argument.Visualise data
p_learn_words <- plot_data(d_learn_words_agg, words_mean, words_se, "Words practiced") +
scale_y_continuous(limits = c(20, 30))
p_learn_words
Fit frequentist model
Prepare data for modelling by mean-centering categorical predictors:
d_learn_words_m <- d_learn_words |>
ungroup() |>
mutate(exp_group_c = ifelse(exp_group == "score", 0, 1),
exp_group_c = exp_group_c - mean(exp_group_c),
gamified_first_c = gamified_first - mean(gamified_first))m_learn_words <- lmer(words_seen ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = d_learn_words_m)
summary(m_learn_words)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
words_seen ~ gamified + gamified:exp_group_c + gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: d_learn_words_m
REML criterion at convergence: 2201.5
Scaled residuals:
Min 1Q Median 3Q Max
-2.22686 -0.49906 0.05179 0.50386 2.35412
Random effects:
Groups Name Variance Std.Dev.
subject (Intercept) 57.91 7.610
Residual 17.11 4.137
Number of obs: 332, groups: subject, 166
Fixed effects:
Estimate Std. Error df
(Intercept) 25.725008 0.672290 203.027802
gamifiedTRUE 0.002888 0.454075 162.000006
gamifiedFALSE:exp_group_c -1.017908 1.346145 203.027801
gamifiedTRUE:exp_group_c -1.655928 1.346145 203.027801
gamifiedFALSE:gamified_first_c 0.317483 1.349388 203.027802
gamifiedTRUE:gamified_first_c -3.005740 1.349388 203.027802
gamifiedFALSE:exp_group_c:gamified_first_c -2.082412 2.702006 203.027802
gamifiedTRUE:exp_group_c:gamified_first_c -4.924329 2.702006 203.027802
t value Pr(>|t|)
(Intercept) 38.265 <2e-16 ***
gamifiedTRUE 0.006 0.9949
gamifiedFALSE:exp_group_c -0.756 0.4504
gamifiedTRUE:exp_group_c -1.230 0.2201
gamifiedFALSE:gamified_first_c 0.235 0.8142
gamifiedTRUE:gamified_first_c -2.227 0.0270 *
gamifiedFALSE:exp_group_c:gamified_first_c -0.771 0.4418
gamifiedTRUE:exp_group_c:gamified_first_c -1.822 0.0699 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) gmTRUE gmfdFALSE:x__ gmfdTRUE:x__ gmfdFALSE:g__
gamifidTRUE -0.338
gmfdFALSE:x__ 0.000 0.000
gmfdTRUE:x__ 0.000 0.000 0.772
gmfdFALSE:g__ 0.000 0.000 -0.004 -0.003
gmfdTRUE:g__ 0.000 0.000 -0.003 -0.004 0.772
gFALSE:__:_ -0.004 0.001 0.000 0.000 0.001
gTRUE:__:__ -0.003 -0.001 0.000 0.000 0.001
gmfdTRUE:g__ gFALSE:__:
gamifidTRUE
gmfdFALSE:x__
gmfdTRUE:x__
gmfdFALSE:g__
gmfdTRUE:g__
gFALSE:__:_ 0.001
gTRUE:__:__ 0.001 0.772 print_model_table(m_learn_words)Effect | Estimate | SE | df | t-value | p-value |
|---|---|---|---|---|---|
Intercept (Control) | 25.725 | 0.672 | 203.03 | 38.26 | <.001 |
Gamification | 0.003 | 0.454 | 162.00 | 0.01 | .995 |
Type of gamification (Points-and-progress - Points) | -1.656 | 1.346 | 203.03 | -1.23 | .220 |
Type of gamification on Control (Points-and-progress - Points) | -1.018 | 1.346 | 203.03 | -0.76 | .450 |
Block on gamification (1 - 2) | -3.006 | 1.349 | 203.03 | -2.23 | .027 |
Block on Control (2 - 1) | 0.317 | 1.349 | 203.03 | 0.24 | .814 |
Type of gamification; block difference (Block 1 - Block 2) | -4.924 | 2.702 | 203.03 | -1.82 | .070 |
Type of gamification on Control; block difference (Block 2 - Block 1) | -2.082 | 2.702 | 203.03 | -0.77 | .442 |
Fitted values
d_model_fit <- crossing(
gamified = TRUE,
exp_group_c = 0,
gamified_first_c = sort(unique(d_learn_words_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_learn_words,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitVisualise fitted model
p_learn_words_m <- plot_model_fit(m_learn_words, d_learn_words_m, y_lab = "Words practiced") +
scale_y_continuous(limits = c(20, 30)) block condition gamified gamified_first exp_group gamified_first_c
1 1 Control FALSE FALSE both -0.5421687
2 1 Control FALSE FALSE score -0.5421687
3 1 Points TRUE TRUE score 0.4578313
4 1 Points-and-progress TRUE TRUE both 0.4578313
5 2 Control FALSE TRUE both 0.4578313
6 2 Control FALSE TRUE score 0.4578313
7 2 Points TRUE FALSE score -0.5421687
8 2 Points-and-progress TRUE FALSE both -0.5421687
exp_group_c pred_val exp_order type
1 0.5240964 25.61111 Control—Both Control
2 -0.4759036 25.50000 Control—Score Control
3 -0.4759036 26.21277 Score—Control Gamified
4 0.5240964 22.30233 Both—Control Gamified
5 0.5240964 24.83721 Both—Control Control
6 -0.4759036 26.80851 Score—Control Control
7 -0.4759036 26.87500 Control—Score Gamified
8 0.5240964 27.88889 Control—Both GamifiedScale for y is already present.
Adding another scale for y, which will replace the existing scale.p_learn_words_m
Fit Bayesian model
As before, we’ll adjust the prior to fit the wider range of likely coefficients, given the scale of the dependent variable.
m_learn_words_bayes <- brm(words_seen ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
family = "gaussian",
data = d_learn_words_m,
prior = set_prior("normal(0, 3)", class = "b"),
chains = 4,
iter = 11000,
warmup = 1000,
sample_prior = TRUE,
future = TRUE,
seed = 0)Compiling Stan program...Trying to compile a simple C fileRunning /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make[1]: *** [foo.o] Error 1Start sampling
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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Chain 4: summary(m_learn_words_bayes) Family: gaussian
Links: mu = identity; sigma = identity
Formula: words_seen ~ gamified + gamified:exp_group_c + gamified:gamified_first_c + gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: d_learn_words_m (Number of observations: 332)
Draws: 4 chains, each with iter = 11000; warmup = 1000; thin = 1;
total post-warmup draws = 40000
Multilevel Hyperparameters:
~subject (Number of levels: 166)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 7.63 0.49 6.72 8.63 1.00 9656 17460
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI
Intercept 25.72 0.68 24.40 27.05
gamifiedTRUE 0.00 0.45 -0.89 0.89
gamifiedFALSE:exp_group_c -0.68 1.17 -2.95 1.64
gamifiedTRUE:exp_group_c -1.29 1.18 -3.59 1.05
gamifiedFALSE:gamified_first_c 0.60 1.17 -1.71 2.90
gamifiedTRUE:gamified_first_c -2.57 1.18 -4.89 -0.26
gamifiedFALSE:exp_group_c:gamified_first_c -0.26 1.84 -3.87 3.33
gamifiedTRUE:exp_group_c:gamified_first_c -2.65 1.84 -6.24 0.94
Rhat Bulk_ESS Tail_ESS
Intercept 1.00 8243 14913
gamifiedTRUE 1.00 66510 28768
gamifiedFALSE:exp_group_c 1.00 10130 17922
gamifiedTRUE:exp_group_c 1.00 9760 17595
gamifiedFALSE:gamified_first_c 1.00 10217 19276
gamifiedTRUE:gamified_first_c 1.00 10360 18651
gamifiedFALSE:exp_group_c:gamified_first_c 1.00 15914 25086
gamifiedTRUE:exp_group_c:gamified_first_c 1.00 15977 26780
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 4.17 0.23 3.74 4.66 1.00 18196 28114
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Inspect the posterior sample distributions of the fixed effects:
plot(m_learn_words_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
Bayes factors
Do a hypothesis learn for all fixed-effect coefficients (both main
effects and interactions) in the model being equal to zero. The column
Evid.Ratio shows the Bayes Factor in favour of the null
hypothesis (\(BF_{01}\)).
h_learn_words <- hypothesis(m_learn_words_bayes,
c("gamifiedTRUE = 0",
"gamifiedFALSE:exp_group_c = 0",
"gamifiedTRUE:exp_group_c = 0",
"gamifiedFALSE:gamified_first_c = 0",
"gamifiedTRUE:gamified_first_c = 0",
"gamifiedFALSE:exp_group_c:gamified_first_c = 0",
"gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
class = "b")
h_learn_words$hypothesis |>
mutate(BF10 = 1 / Evid.Ratio,
evidence_for_null = sapply(Evid.Ratio, bf_to_strength))This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line). Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.
sd_ratio_words <- plot(h_learn_words, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)
sd_ratio_words[[1]] +
geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
Completed trials
Prepare data
d_learn_trials <- d_learn |>
group_by(subject, exp_group, block, condition, gamified, gamified_first) |>
summarise(n_trials = n())`summarise()` has grouped output by 'subject', 'exp_group', 'block',
'condition', 'gamified'. You can override using the `.groups` argument.d_learn_trials_agg <- d_learn_trials |>
group_by(block, condition, gamified, gamified_first, exp_group) |>
summarise(trials_mean = mean(n_trials, na.rm = T),
trials_se = sd(n_trials, na.rm = T)/sqrt(n())) |>
ungroup() |>
add_experiment_cols()`summarise()` has grouped output by 'block', 'condition', 'gamified',
'gamified_first'. You can override using the `.groups` argument.Visualise data
p_learn_trials <- plot_data(d_learn_trials_agg, trials_mean, trials_se, "Completed trials") +
scale_y_continuous(limits = c(130, 170))
p_learn_trials
Fit frequentist model
Prepare data for modelling by mean-centering categorical predictors:
d_learn_trials_m <- d_learn_trials |>
ungroup() |>
mutate(exp_group_c = ifelse(exp_group == "score", 0, 1),
exp_group_c = exp_group_c - mean(exp_group_c),
gamified_first_c = gamified_first - mean(gamified_first))m_learn_trials <- lmer(n_trials ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = d_learn_trials_m)
summary(m_learn_trials)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
n_trials ~ gamified + gamified:exp_group_c + gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: d_learn_trials_m
REML criterion at convergence: 2932.7
Scaled residuals:
Min 1Q Median 3Q Max
-2.34679 -0.49538 -0.03198 0.45220 2.45632
Random effects:
Groups Name Variance Std.Dev.
subject (Intercept) 586.6 24.22
Residual 156.2 12.50
Number of obs: 332, groups: subject, 166
Fixed effects:
Estimate Std. Error df t value
(Intercept) 151.1115 2.1153 199.5462 71.439
gamifiedTRUE -0.8356 1.3717 162.0000 -0.609
gamifiedFALSE:exp_group_c -3.6333 4.2355 199.5462 -0.858
gamifiedTRUE:exp_group_c -4.5968 4.2355 199.5462 -1.085
gamifiedFALSE:gamified_first_c -0.6598 4.2457 199.5462 -0.155
gamifiedTRUE:gamified_first_c -12.1755 4.2457 199.5462 -2.868
gamifiedFALSE:exp_group_c:gamified_first_c -2.9971 8.5015 199.5462 -0.353
gamifiedTRUE:exp_group_c:gamified_first_c -4.7077 8.5015 199.5462 -0.554
Pr(>|t|)
(Intercept) < 2e-16 ***
gamifiedTRUE 0.54326
gamifiedFALSE:exp_group_c 0.39202
gamifiedTRUE:exp_group_c 0.27909
gamifiedFALSE:gamified_first_c 0.87665
gamifiedTRUE:gamified_first_c 0.00458 **
gamifiedFALSE:exp_group_c:gamified_first_c 0.72481
gamifiedTRUE:exp_group_c:gamified_first_c 0.58037
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) gmTRUE gmfdFALSE:x__ gmfdTRUE:x__ gmfdFALSE:g__
gamifidTRUE -0.324
gmfdFALSE:x__ 0.000 0.000
gmfdTRUE:x__ 0.000 0.000 0.790
gmfdFALSE:g__ 0.000 0.000 -0.004 -0.003
gmfdTRUE:g__ 0.000 0.000 -0.003 -0.004 0.790
gFALSE:__:_ -0.004 0.001 0.000 0.000 0.001
gTRUE:__:__ -0.003 -0.001 0.000 0.000 0.001
gmfdTRUE:g__ gFALSE:__:
gamifidTRUE
gmfdFALSE:x__
gmfdTRUE:x__
gmfdFALSE:g__
gmfdTRUE:g__
gFALSE:__:_ 0.001
gTRUE:__:__ 0.001 0.790 print_model_table(m_learn_trials)Effect | Estimate | SE | df | t-value | p-value |
|---|---|---|---|---|---|
Intercept (Control) | 151.111 | 2.115 | 199.55 | 71.44 | <.001 |
Gamification | -0.836 | 1.372 | 162.00 | -0.61 | .543 |
Type of gamification (Points-and-progress - Points) | -4.597 | 4.235 | 199.55 | -1.09 | .279 |
Type of gamification on Control (Points-and-progress - Points) | -3.633 | 4.235 | 199.55 | -0.86 | .392 |
Block on gamification (1 - 2) | -12.175 | 4.246 | 199.55 | -2.87 | .005 |
Block on Control (2 - 1) | -0.660 | 4.246 | 199.55 | -0.16 | .877 |
Type of gamification; block difference (Block 1 - Block 2) | -4.708 | 8.501 | 199.55 | -0.55 | .580 |
Type of gamification on Control; block difference (Block 2 - Block 1) | -2.997 | 8.501 | 199.55 | -0.35 | .725 |
Fitted values
d_model_fit <- crossing(
gamified = TRUE,
exp_group_c = 0,
gamified_first_c = sort(unique(d_learn_trials_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_learn_trials,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitVisualise fitted model
p_learn_trials_m <- plot_model_fit(m_learn_trials, d_learn_trials_m, y_lab = "Trials completed") +
scale_y_continuous(limits = c(130, 170)) block condition gamified gamified_first exp_group gamified_first_c
1 1 Control FALSE FALSE both -0.5421687
2 1 Control FALSE FALSE score -0.5421687
3 1 Points TRUE TRUE score 0.4578313
4 1 Points-and-progress TRUE TRUE both 0.4578313
5 2 Control FALSE TRUE both 0.4578313
6 2 Control FALSE TRUE score 0.4578313
7 2 Points TRUE FALSE score -0.5421687
8 2 Points-and-progress TRUE FALSE both -0.5421687
exp_group_c pred_val exp_order type
1 0.5240964 150.4167 Control—Both Control
2 -0.4759036 152.4250 Control—Score Control
3 -0.4759036 147.9149 Score—Control Gamified
4 0.5240964 141.1628 Both—Control Gamified
5 0.5240964 148.1860 Both—Control Control
6 -0.4759036 153.1915 Score—Control Control
7 -0.4759036 157.8500 Control—Score Gamified
8 0.5240964 155.8056 Control—Both GamifiedScale for y is already present.
Adding another scale for y, which will replace the existing scale.p_learn_trials_m
Fit Bayesian model
m_learn_trials_bayes <- brm(n_trials ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
family = "gaussian",
data = d_learn_trials_m,
prior = set_prior("normal(0, 10)", class = "b"),
chains = 4,
iter = 11000,
warmup = 1000,
sample_prior = TRUE,
future = TRUE,
seed = 0)Compiling Stan program...Trying to compile a simple C fileRunning /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make[1]: *** [foo.o] Error 1Start sampling
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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Chain 4: summary(m_learn_trials_bayes) Family: gaussian
Links: mu = identity; sigma = identity
Formula: n_trials ~ gamified + gamified:exp_group_c + gamified:gamified_first_c + gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: d_learn_trials_m (Number of observations: 332)
Draws: 4 chains, each with iter = 11000; warmup = 1000; thin = 1;
total post-warmup draws = 40000
Multilevel Hyperparameters:
~subject (Number of levels: 166)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 24.23 1.55 21.36 27.44 1.00 9301 17644
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI
Intercept 151.10 2.13 146.91 155.30
gamifiedTRUE -0.81 1.37 -3.53 1.90
gamifiedFALSE:exp_group_c -2.69 3.71 -9.90 4.58
gamifiedTRUE:exp_group_c -3.62 3.72 -10.90 3.65
gamifiedFALSE:gamified_first_c 0.73 3.73 -6.64 8.05
gamifiedTRUE:gamified_first_c -10.34 3.73 -17.69 -2.96
gamifiedFALSE:exp_group_c:gamified_first_c -0.95 5.92 -12.59 10.69
gamifiedTRUE:exp_group_c:gamified_first_c -2.43 5.91 -13.96 9.08
Rhat Bulk_ESS Tail_ESS
Intercept 1.00 6797 13747
gamifiedTRUE 1.00 68590 27470
gamifiedFALSE:exp_group_c 1.00 8772 17479
gamifiedTRUE:exp_group_c 1.00 8995 17343
gamifiedFALSE:gamified_first_c 1.00 8900 16298
gamifiedTRUE:gamified_first_c 1.00 8886 16215
gamifiedFALSE:exp_group_c:gamified_first_c 1.00 13869 21955
gamifiedTRUE:exp_group_c:gamified_first_c 1.00 13753 21771
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 12.59 0.71 11.30 14.08 1.00 18091 25582
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Inspect the posterior sample distributions of the fixed effects:
plot(m_learn_trials_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
Bayes factors
Do a hypothesis learn for all fixed-effect coefficients (both main
effects and interactions) in the model being equal to zero. The column
Evid.Ratio shows the Bayes Factor in favour of the null
hypothesis (\(BF_{01}\)).
h_learn_trials <- hypothesis(m_learn_trials_bayes,
c("gamifiedTRUE = 0",
"gamifiedFALSE:exp_group_c = 0",
"gamifiedTRUE:exp_group_c = 0",
"gamifiedFALSE:gamified_first_c = 0",
"gamifiedTRUE:gamified_first_c = 0",
"gamifiedFALSE:exp_group_c:gamified_first_c = 0",
"gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
class = "b")
h_learn_trials$hypothesis |>
mutate(BF10 = 1 / Evid.Ratio,
evidence_for_null = sapply(Evid.Ratio, bf_to_strength))This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line). Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.
sd_ratio_trials <- plot(h_learn_trials, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)
sd_ratio_trials[[1]] +
geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
Conclusions
- Gamified feedback had no effect on response accuracy during practice. Response accuracy was slightly higher in the Control condition for participants in the Points group than for participants in the Progress bar group, possibly due to group differences in spite of random assignment.
- Correct responses were slower during practice with gamified feedback than in the control condition, particularly so in Block 1.
- There were no overall effects of the gamification manipulation on total score, number of words practiced, or number of trials completed. However, in the gamified conditions, these outcomes were all better in Block 2 than in Block 1.
Combined plot
((p_learn_acc | p_learn_rt) / (p_learn_trials | p_learn_words | p_learn_score)) +
plot_layout(guides = "collect") +
plot_annotation(tag_levels = "a")
ggsave(here("output", "practice_performance_all.png"), width = 7.5, height = 5)Streamlined version:
(p_learn_acc | p_learn_rt | p_learn_score) +
plot_layout(guides = "collect") +
plot_annotation(tag_levels = "a")
ggsave(here("output", "practice_performance.png"), width = 7.5, height = 3)Save hypothesis testing output for visualisation
fwrite(h_learn_acc$hypothesis, here("output", "hypothesis_tests", "h_learn_acc.csv"))
fwrite(h_learn_rt$hypothesis, here("output", "hypothesis_tests", "h_learn_rt.csv"))
fwrite(h_learn_score$hypothesis, here("output", "hypothesis_tests", "h_learn_score.csv"))
fwrite(h_learn_words$hypothesis, here("output", "hypothesis_tests", "h_learn_words.csv"))
fwrite(h_learn_trials$hypothesis, here("output", "hypothesis_tests", "h_learn_trials.csv"))Session info
sessionInfo()R version 4.3.1 (2023-06-16)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Sonoma 14.5
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
time zone: Europe/Amsterdam
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] flextable_0.9.6 data.table_1.14.8 future_1.33.0 brms_2.21.0
[5] Rcpp_1.0.11 lmerTest_3.1-3 lme4_1.1-34 Matrix_1.6-1.1
[9] tidyr_1.3.0 stringr_1.5.0 patchwork_1.3.0 scales_1.3.0
[13] ggplot2_3.5.0 dplyr_1.1.3 here_1.0.1
loaded via a namespace (and not attached):
[1] gridExtra_2.3 inline_0.3.19 sandwich_3.0-2
[4] rlang_1.1.1 magrittr_2.0.3 multcomp_1.4-25
[7] matrixStats_1.3.0 compiler_4.3.1 loo_2.7.0
[10] reshape2_1.4.4 systemfonts_1.0.4 callr_3.7.3
[13] vctrs_0.6.3 httpcode_0.3.0 pkgconfig_2.0.3
[16] crayon_1.5.2 fastmap_1.1.1 ellipsis_0.3.2
[19] backports_1.4.1 labeling_0.4.3 utf8_1.2.3
[22] promises_1.2.1 rmarkdown_2.25 ps_1.7.5
[25] nloptr_2.0.3 ragg_1.2.7 purrr_1.0.2
[28] xfun_0.40 cachem_1.0.8 jsonlite_1.8.7
[31] later_1.3.1 uuid_1.2-0 parallel_4.3.1
[34] prettyunits_1.2.0 R6_2.5.1 bslib_0.5.1
[37] stringi_1.7.12 StanHeaders_2.32.6 parallelly_1.36.0
[40] boot_1.3-28.1 jquerylib_0.1.4 numDeriv_2016.8-1.1
[43] estimability_1.4.1 rstan_2.32.6 knitr_1.44
[46] zoo_1.8-12 bayesplot_1.11.1 httpuv_1.6.12
[49] splines_4.3.1 tidyselect_1.2.0 rstudioapi_0.15.0
[52] abind_1.4-5 yaml_2.3.7 codetools_0.2-19
[55] curl_5.1.0 processx_3.8.2 listenv_0.9.0
[58] pkgbuild_1.4.2 plyr_1.8.9 lattice_0.21-9
[61] tibble_3.2.1 shiny_1.8.0 withr_2.5.1
[64] bridgesampling_1.1-2 askpass_1.2.0 posterior_1.5.0
[67] coda_0.19-4 evaluate_0.22 survival_3.5-7
[70] RcppParallel_5.1.7 zip_2.3.1 xml2_1.3.5
[73] pillar_1.9.0 tensorA_0.36.2.1 checkmate_2.3.1
[76] stats4_4.3.1 distributional_0.4.0 generics_0.1.3
[79] rprojroot_2.0.3 rstantools_2.4.0 munsell_0.5.0
[82] minqa_1.2.6 globals_0.16.2 xtable_1.8-4
[85] glue_1.6.2 gdtools_0.3.7 emmeans_1.8.9
[88] tools_4.3.1 gfonts_0.2.0 mvtnorm_1.2-3
[91] grid_4.3.1 QuickJSR_1.1.3 colorspace_2.1-0
[94] nlme_3.1-163 cli_3.6.1 textshaping_0.3.7
[97] officer_0.6.6 fontBitstreamVera_0.1.1 fansi_1.0.4
[100] Brobdingnag_1.2-9 V8_4.3.3 gtable_0.3.4
[103] sass_0.4.7 digest_0.6.33 fontquiver_0.2.1
[106] crul_1.4.2 TH.data_1.1-2 farver_2.1.1
[109] htmltools_0.5.6 lifecycle_1.0.3 mime_0.12
[112] openssl_2.1.1 fontLiberation_0.1.0 MASS_7.3-60 ---
title: "Analysis: learning session" 
subtitle: "Gamified Feedback in Adaptive Retrieval Practice"
author: "Maarten van der Velde & Gesa van den Broek"
date: "Last updated: `r Sys.Date()`"
output:
  html_notebook:
    smart: no
    toc: yes
    toc_float: yes
  github_document:
    toc: yes
editor_options: 
  chunk_output_type: inline
---

```{r setup, include = FALSE}
# Remove to disable caching
knitr::opts_chunk$set(cache = TRUE)
```

# Setup

```{r}
library(here)
library(dplyr)
library(ggplot2)
library(scales)
library(patchwork)
library(stringr)
library(tidyr)
library(lme4)
library(lmerTest)
library(brms)

# Set up parallel processing for Bayesian models
library(future)
plan(multisession, workers = 4)
```

Helper functions for plots and tables:
```{r}
source(here("scripts", "00_visualisation_functions.R"))
```

Load processed data:
```{r}
d_learn <- readRDS(here("data", "processed", "d_learn.rds"))
```

```{r}
add_experiment_cols <- function (data) {
  data |>
    mutate(exp_order = case_when(
      gamified_first == 0 & exp_group == "score" ~ "Control—Score",
      gamified_first == 0 & exp_group == "both" ~ "Control—Both",
      gamified_first == 1 & exp_group == "score" ~ "Score—Control",
      gamified_first == 1 & exp_group == "both" ~ "Both—Control"
    )) |>
    mutate(type = ifelse(gamified, "Gamified", "Control"))
}
```

Helper function for interpreting Bayes factors:
```{r}
bf_to_strength <- function (bf) {
  
  direction <- "for"
  
  if (bf < 1) {
    bf <- 1/bf
    direction <- "against"
  }
  
  strength <- case_when(
    bf == 1 ~ "No",
    bf < 3 ~ "Anecdotal",
    bf < 10 ~ "Moderate",
    bf < 30 ~ "Strong",
    bf < 100 ~ "Very strong",
    TRUE ~ "Extreme"
  )
  
  paste0(strength, " evidence ", direction)
}
```

# Does gamification change performance during practice?

## Accuracy

### Prepare data
```{r}
d_learn_acc <- d_learn |>
  filter(!study_trial) |>
  group_by(subject, exp_group, block, condition, gamified, gamified_first) |>
  summarise(accuracy = mean(correct))

d_learn_acc_agg <- d_learn_acc |>
  group_by(block, condition, gamified, gamified_first, exp_group) |>
  summarise(acc = mean(accuracy, na.rm = T),
            acc_se = sd(accuracy, na.rm = T)/sqrt(n())) |>
  ungroup() |>
  add_experiment_cols()
```

### Visualise data

```{r}
p_learn_acc <- plot_data(d_learn_acc_agg, acc, acc_se, "Accuracy") +
  scale_y_continuous(limits = c(.725, .875), labels = scales::percent_format())

p_learn_acc
```

### Fit frequentist model

Prepare data for modelling by mean-centering categorical predictors:
```{r}
d_learn_m <- d_learn |>
  filter(!study_trial) |>
  mutate(exp_group_c = ifelse(exp_group == "score", 0, 1),
         exp_group_c = exp_group_c - mean(exp_group_c),
         gamified_first_c = gamified_first - mean(gamified_first))
```

```{r}
m_learn_acc <- glmer(correct ~ gamified +
                       gamified:exp_group_c +
                       gamified:gamified_first_c +
                       gamified:gamified_first_c:exp_group_c +
                       (1 | subject) + (1 | fact),
                     family = "binomial",
                     data = d_learn_m)

summary(m_learn_acc)
print_model_table(m_learn_acc)
```


#### Fitted values

```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = sort(unique(d_learn_m$exp_group_c)), 
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_learn_acc,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```



#### Visualise fitted model

```{r}
p_learn_acc_m <- plot_model_fit(m_learn_acc, d_learn_m, y_lab = "Accuracy") +
  scale_y_continuous(limits = c(.75, .90), labels = scales::percent_format(accuracy = .1))

p_learn_acc_m
```
### Fit Bayesian model

Fit again with `brms` so that we can calculate Bayes Factors.
Because we expect any fixed effects to be at most moderate in size, we will use a weakly informative Normal(0, 1) prior for these effects.
```{r}
m_learn_acc_brms <- brm(correct ~ gamified +
                          gamified:exp_group_c +
                          gamified:gamified_first_c +
                          gamified:gamified_first_c:exp_group_c +
                          (1 | subject) + (1 | fact),
                        family = "bernoulli",
                        data = d_learn_m,
                        prior = set_prior("normal(0, 1)", class = "b"),
                        chains = 4,
                        iter = 11000,
                        warmup = 1000,
                        sample_prior = TRUE,
                        future = TRUE,
                        seed = 0)

summary(m_learn_acc_brms)
```

Inspect the posterior sample distributions of the fixed effects:
```{r, fig.height = 12, fig.width = 8}
plot(m_learn_acc_brms, nvariables = 8, variable = "^b_", regex = TRUE)
```

#### Bayes factors

Do a hypothesis test for all fixed-effect coefficients (both main effects and interactions) in the model being equal to zero.
The column `Evid.Ratio` shows the Bayes Factor in favour of the null hypothesis ($BF_{01}$).
```{r}
h_learn_acc <- hypothesis(m_learn_acc_brms,
                          c("gamifiedTRUE = 0",
                            "gamifiedFALSE:exp_group_c = 0",
                            "gamifiedTRUE:exp_group_c = 0",
                            "gamifiedFALSE:gamified_first_c = 0",
                            "gamifiedTRUE:gamified_first_c = 0",
                            "gamifiedFALSE:exp_group_c:gamified_first_c = 0",
                            "gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
                          class = "b")

h_learn_acc$hypothesis |>
  mutate(BF10 = 1 / Evid.Ratio,
         evidence_for_null = sapply(Evid.Ratio, bf_to_strength))
```

This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line).
Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.  
```{r, fig.height = 12, fig.width = 6}
sd_ratio_acc <- plot(h_learn_acc, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)

sd_ratio_acc[[1]] +
  geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
```



#### Conclusion

The Bayesian model finds anecdotal to strong evidence in favour of the null hypothesis (no effect on learning accuracy) for each of the regression coefficients.

## Response time

Response time on correct answers only.

### Prepare data

To keep the visualisation of average response times by condition simple, we calculate the median RT per participant, and then take the mean and SD of these medians (which are themselves roughly normally distributed).
```{r}
d_learn_rt <- d_learn |>
  filter(!study_trial) |>
  filter(correct) |>
  mutate(rt = rt / 1000) |>
  group_by(subject, exp_group, block, condition, gamified, gamified_first) |>
  summarise(rt_median = median(rt, na.rm = TRUE))

d_learn_rt_agg <- d_learn_rt |>
  group_by(block, condition, gamified, gamified_first, exp_group) |>
  summarise(rt_mean = mean(rt_median, na.rm = T),
            rt_se = sd(rt_median, na.rm = T)/sqrt(n())) |>
  ungroup() |>
  add_experiment_cols()
```

### Visualise data

```{r}
p_learn_rt <- plot_data(d_learn_rt_agg, rt_mean, rt_se, "Response time (s)") +
  scale_y_continuous(limits = c(1.3, 1.8), labels = scales::comma_format())

p_learn_rt
```


### Fit frequentist model

Since RT data is not normally distributed, we fit a lognormal model to the response times. 
(See https://bbolker.github.io/mixedmodels-misc/glmmFAQ.html#gamma-glmms .)
Prepare data for modelling by mean-centering categorical predictors:
```{r}
d_learn_rt_m <- d_learn |>
  filter(!study_trial) |>
  filter(correct) |>
  mutate(log_rt = log(rt / 1000)) |>
  mutate(exp_group_c = ifelse(exp_group == "score", 0, 1),
         exp_group_c = exp_group_c - mean(exp_group_c),
         gamified_first_c = gamified_first - mean(gamified_first)
         )
```


```{r}
m_learn_rt <- lmer(log_rt ~ gamified +
                      gamified:exp_group_c +
                      gamified:gamified_first_c +
                      gamified:gamified_first_c:exp_group_c +
                      (1 | subject) + (1 | fact),
                    data = d_learn_rt_m)

summary(m_learn_rt)
print_model_table(m_learn_rt)
```


#### Fitted values
```{r}
d_model_fit <- crossing(
  gamified = c(FALSE, TRUE), 
  exp_group_c = 0, 
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_learn_rt,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response") |>
  exp() # Transform logRT to RT

d_model_fit
```

```{r}
d_model_fit <- crossing(
  gamified = c(FALSE, TRUE), 
  exp_group_c = 0, 
  gamified_first_c = sort(unique(d_learn_rt_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_learn_rt,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response") |>
  exp() # Transform logRT to RT

d_model_fit
```

### Visualise fitted model

```{r}
p_learn_rt_m <- plot_model_fit(m_learn_rt, d_learn_rt_m, exp_trans = TRUE, y_lab = "Response time (s)") +
  scale_y_continuous(limits = c(1.4, 1.9), labels = scales::comma_format())

p_learn_rt_m
```
### Fit Bayesian model

Fit again with `brms` so that we can calculate Bayes Factors.
Because we expect any fixed effects to be at most moderate in size, we will use a weakly informative Normal(0, .1) prior for these effects.
```{r}
m_learn_rt_brms <- brm(log_rt ~ gamified +
                         gamified:exp_group_c +
                         gamified:gamified_first_c +
                         gamified:gamified_first_c:exp_group_c +
                         (1 | subject) + (1 | fact),
                       family = "gaussian",
                       data = d_learn_rt_m,
                       prior = set_prior("normal(0, .1)", class = "b"),
                       chains = 4,
                       iter = 11000,
                       warmup = 1000,
                       sample_prior = TRUE,
                       future = TRUE,
                       seed = 0)

summary(m_learn_rt_brms)
```

Inspect the posterior sample distributions of the fixed effects:
```{r, fig.height = 12, fig.width = 8}
plot(m_learn_rt_brms, nvariables = 8, variable = "^b_", regex = TRUE)
```


#### Bayes factors

Do a hypothesis learn for all fixed-effect coefficients (both main effects and interactions) in the model being equal to zero.
The column `Evid.Ratio` shows the Bayes Factor in favour of the null hypothesis ($BF_{01}$).
```{r}
h_learn_rt <- hypothesis(m_learn_rt_brms,
                         c("gamifiedTRUE = 0",
                           "gamifiedFALSE:exp_group_c = 0",
                           "gamifiedTRUE:exp_group_c = 0",
                           "gamifiedFALSE:gamified_first_c = 0",
                           "gamifiedTRUE:gamified_first_c = 0",
                           "gamifiedFALSE:exp_group_c:gamified_first_c = 0",
                           "gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
                         class = "b")


h_learn_rt$hypothesis |>
  mutate(BF10 = 1 / Evid.Ratio,
         evidence_for_null = sapply(Evid.Ratio, bf_to_strength))
```

This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line).
Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.  
```{r, fig.height = 12, fig.width = 6}
sd_ratio_rt <- plot(h_learn_rt, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)

sd_ratio_rt[[1]] +
  geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
```

#### Fitted values
```{r}
d_model_fit <- crossing(
  gamified = c(FALSE, TRUE), 
  exp_group_c = 0, 
  gamified_first_c = sort(unique(d_learn_rt_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_learn_rt_brms,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1] |>
  exp() # Transform logRT to RT

d_model_fit
```


#### Conclusion
The Bayesian model finds anecdotal to strong evidence in favour of the null hypothesis (no effect on correct RT) for each of the regression coefficients.

## Total score

The total score is the number of points after the last trial in a block.

### Prepare data
```{r}
d_learn_score <- d_learn |>
  group_by(subject, exp_group, block, condition, gamified, gamified_first) |>
  slice(n())

d_learn_score_agg <- d_learn_score |>
  group_by(block, condition, gamified, gamified_first, exp_group) |>
  summarise(feedback_score_mean = mean(feedback_score, na.rm = T),
            feedback_score_se = sd(feedback_score, na.rm = T)/sqrt(n())) |>
  ungroup() |>
  add_experiment_cols()
```

### Visualise data

```{r}
p_learn_score <- plot_data(d_learn_score_agg, feedback_score_mean, feedback_score_se, "Total score") +
  scale_y_continuous(limits = c(1000, 1400), labels = scales::comma_format())

p_learn_score
```


Distribution of scores:
```{r}
p_learn_score_dist <- ggplot(d_learn_score, aes(x = feedback_score, fill = condition)) +
  facet_grid(condition ~ .) +
  geom_histogram(aes(y=..density..), colour = "black", binwidth = 100) +
  geom_density(alpha = .5) +
  geom_vline(xintercept = c(1200, 1500), lty = 2) +
  scale_fill_manual(values = col_condition) +
  scale_colour_manual(values = col_condition) +
  guides(fill = "none",
         colour = "none") +
  labs(x = "Total score",
       y = "Density") +
  theme_paper

p_learn_score_dist

ggsave(here("output", "practice_scores.png"), width = 7.5, height = 5)
```


### Fit frequentist model

Prepare data for modelling by mean-centering categorical predictors:
```{r}
d_learn_score_m <- d_learn_score |>
  ungroup() |>
  mutate(exp_group_c = ifelse(exp_group == "score", 0, 1),
         exp_group_c = exp_group_c - mean(exp_group_c),
         gamified_first_c = gamified_first - mean(gamified_first))
```


```{r}
m_learn_score <- lmer(feedback_score ~ gamified +
                       gamified:exp_group_c +
                       gamified:gamified_first_c +
                       gamified:gamified_first_c:exp_group_c +
                       (1 | subject),
                     data = d_learn_score_m)

summary(m_learn_score)
print_model_table(m_learn_score)
```


#### Fitted values

```{r}
d_model_fit <- crossing(
  gamified = TRUE, 
  exp_group_c = 0,
  gamified_first_c = sort(unique(d_learn_score_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_learn_score,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```

### Fit Bayesian model

Since score is on a much larger scale than other dependent variables, we expect coefficients for  fixed effects to be on a much larger scale too.
Adjust the prior accordingly:
```{r}
m_learn_score_bayes <- brm(feedback_score ~ gamified +
                             gamified:exp_group_c +
                             gamified:gamified_first_c +
                             gamified:gamified_first_c:exp_group_c +
                             (1 | subject),
                           family = "gaussian",
                           data = d_learn_score_m,
                           prior = set_prior("normal(0, 100)", class = "b"),
                           chains = 4,
                           iter = 11000,
                           warmup = 1000,
                           sample_prior = TRUE,
                           future = TRUE,
                           seed = 0)

summary(m_learn_score_bayes)
```

Inspect the posterior sample distributions of the fixed effects:
```{r, fig.height = 12, fig.width = 8}
plot(m_learn_score_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
```


#### Bayes factors

Do a hypothesis learn for all fixed-effect coefficients (both main effects and interactions) in the model being equal to zero.
The column `Evid.Ratio` shows the Bayes Factor in favour of the null hypothesis ($BF_{01}$).
```{r}
h_learn_score <- hypothesis(m_learn_score_bayes,
                            c("gamifiedTRUE = 0",
                              "gamifiedFALSE:exp_group_c = 0",
                              "gamifiedTRUE:exp_group_c = 0",
                              "gamifiedFALSE:gamified_first_c = 0",
                              "gamifiedTRUE:gamified_first_c = 0",
                              "gamifiedFALSE:exp_group_c:gamified_first_c = 0",
                              "gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
                            class = "b")


h_learn_score$hypothesis |>
  mutate(BF10 = 1 / Evid.Ratio,
         evidence_for_null = sapply(Evid.Ratio, bf_to_strength))
```

This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line).
Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.  
```{r, fig.height = 12, fig.width = 6}
sd_ratio_score <- plot(h_learn_score, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)

sd_ratio_score[[1]] +
  geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
```

#### Fitted values

```{r}
d_model_fit <- crossing(
  gamified = TRUE, 
  exp_group_c = 0,
  gamified_first_c = sort(unique(d_learn_score_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_learn_score_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```



## Number of words practiced

### Prepare data
```{r}
d_learn_words <- d_learn |>
  group_by(subject, exp_group, block, condition, gamified, gamified_first) |>
  summarise(words_seen = n_distinct(fact))

d_learn_words_agg <- d_learn_words |>
  group_by(block, condition, gamified, gamified_first, exp_group) |>
  summarise(words_mean = mean(words_seen, na.rm = T),
            words_se = sd(words_seen, na.rm = T)/sqrt(n())) |>
  ungroup() |>
  add_experiment_cols()
```

### Visualise data

```{r}
p_learn_words <- plot_data(d_learn_words_agg, words_mean, words_se, "Words practiced") +
  scale_y_continuous(limits = c(20, 30))

p_learn_words
```


### Fit frequentist model

Prepare data for modelling by mean-centering categorical predictors:
```{r}
d_learn_words_m <- d_learn_words |>
  ungroup() |>
  mutate(exp_group_c = ifelse(exp_group == "score", 0, 1),
         exp_group_c = exp_group_c - mean(exp_group_c),
         gamified_first_c = gamified_first - mean(gamified_first))
```


```{r}
m_learn_words <- lmer(words_seen ~ gamified +
                       gamified:exp_group_c +
                       gamified:gamified_first_c +
                       gamified:gamified_first_c:exp_group_c +
                       (1 | subject),
                     data = d_learn_words_m)

summary(m_learn_words)
print_model_table(m_learn_words)
```


#### Fitted values
```{r}
d_model_fit <- crossing(
  gamified = TRUE, 
  exp_group_c = 0,
  gamified_first_c = sort(unique(d_learn_words_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_learn_words,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```


#### Visualise fitted model
```{r}
p_learn_words_m <- plot_model_fit(m_learn_words, d_learn_words_m, y_lab = "Words practiced") +
  scale_y_continuous(limits = c(20, 30))

p_learn_words_m
```

### Fit Bayesian model

As before, we'll adjust the prior to fit the wider range of likely coefficients, given the scale of the dependent variable.
```{r}
m_learn_words_bayes <- brm(words_seen ~ gamified +
                           gamified:exp_group_c +
                           gamified:gamified_first_c +
                           gamified:gamified_first_c:exp_group_c +
                           (1 | subject),
                         family = "gaussian",
                         data = d_learn_words_m,
                         prior = set_prior("normal(0, 3)", class = "b"),
                         chains = 4,
                         iter = 11000,
                         warmup = 1000,
                         sample_prior = TRUE,
                         future = TRUE,
                         seed = 0)

summary(m_learn_words_bayes)
```

Inspect the posterior sample distributions of the fixed effects:
```{r, fig.height = 12, fig.width = 8}
plot(m_learn_words_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
```


#### Bayes factors

Do a hypothesis learn for all fixed-effect coefficients (both main effects and interactions) in the model being equal to zero.
The column `Evid.Ratio` shows the Bayes Factor in favour of the null hypothesis ($BF_{01}$).
```{r}
h_learn_words <- hypothesis(m_learn_words_bayes,
                            c("gamifiedTRUE = 0",
                              "gamifiedFALSE:exp_group_c = 0",
                              "gamifiedTRUE:exp_group_c = 0",
                              "gamifiedFALSE:gamified_first_c = 0",
                              "gamifiedTRUE:gamified_first_c = 0",
                              "gamifiedFALSE:exp_group_c:gamified_first_c = 0",
                              "gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
                            class = "b")


h_learn_words$hypothesis |>
  mutate(BF10 = 1 / Evid.Ratio,
         evidence_for_null = sapply(Evid.Ratio, bf_to_strength))
```

This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line).
Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.  
```{r, fig.height = 12, fig.width = 6}
sd_ratio_words <- plot(h_learn_words, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)

sd_ratio_words[[1]] +
  geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
```

## Completed trials

### Prepare data
```{r}
d_learn_trials <- d_learn |>
  group_by(subject, exp_group, block, condition, gamified, gamified_first) |>
  summarise(n_trials = n())

d_learn_trials_agg <- d_learn_trials |>
  group_by(block, condition, gamified, gamified_first, exp_group) |>
  summarise(trials_mean = mean(n_trials, na.rm = T),
            trials_se = sd(n_trials, na.rm = T)/sqrt(n())) |>
  ungroup() |>
  add_experiment_cols()
```

### Visualise data

```{r}
p_learn_trials <- plot_data(d_learn_trials_agg, trials_mean, trials_se, "Completed trials") +
  scale_y_continuous(limits = c(130, 170))

p_learn_trials
```



### Fit frequentist model

Prepare data for modelling by mean-centering categorical predictors:
```{r}
d_learn_trials_m <- d_learn_trials |>
  ungroup() |>
  mutate(exp_group_c = ifelse(exp_group == "score", 0, 1),
         exp_group_c = exp_group_c - mean(exp_group_c),
         gamified_first_c = gamified_first - mean(gamified_first))
```


```{r}
m_learn_trials <- lmer(n_trials ~ gamified +
                       gamified:exp_group_c +
                       gamified:gamified_first_c +
                       gamified:gamified_first_c:exp_group_c +
                       (1 | subject),
                     data = d_learn_trials_m)

summary(m_learn_trials)
print_model_table(m_learn_trials)
```


#### Fitted values
```{r}
d_model_fit <- crossing(
  gamified = TRUE, 
  exp_group_c = 0,
  gamified_first_c = sort(unique(d_learn_trials_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_learn_trials,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```


#### Visualise fitted model
```{r}
p_learn_trials_m <- plot_model_fit(m_learn_trials, d_learn_trials_m, y_lab = "Trials completed") +
  scale_y_continuous(limits = c(130, 170))

p_learn_trials_m
```

### Fit Bayesian model

```{r}
m_learn_trials_bayes <- brm(n_trials ~ gamified +
                           gamified:exp_group_c +
                           gamified:gamified_first_c +
                           gamified:gamified_first_c:exp_group_c +
                           (1 | subject),
                         family = "gaussian",
                         data = d_learn_trials_m,
                         prior = set_prior("normal(0, 10)", class = "b"),
                         chains = 4,
                         iter = 11000,
                         warmup = 1000,
                         sample_prior = TRUE,
                         future = TRUE,
                         seed = 0)

summary(m_learn_trials_bayes)
```

Inspect the posterior sample distributions of the fixed effects:
```{r, fig.height = 12, fig.width = 8}
plot(m_learn_trials_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
```


#### Bayes factors

Do a hypothesis learn for all fixed-effect coefficients (both main effects and interactions) in the model being equal to zero.
The column `Evid.Ratio` shows the Bayes Factor in favour of the null hypothesis ($BF_{01}$).
```{r}
h_learn_trials <- hypothesis(m_learn_trials_bayes,
                             c("gamifiedTRUE = 0",
                               "gamifiedFALSE:exp_group_c = 0",
                               "gamifiedTRUE:exp_group_c = 0",
                               "gamifiedFALSE:gamified_first_c = 0",
                               "gamifiedTRUE:gamified_first_c = 0",
                               "gamifiedFALSE:exp_group_c:gamified_first_c = 0",
                               "gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
                             class = "b")


h_learn_trials$hypothesis |>
  mutate(BF10 = 1 / Evid.Ratio,
         evidence_for_null = sapply(Evid.Ratio, bf_to_strength))
```

This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line).
Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.  
```{r, fig.height = 12, fig.width = 6}
sd_ratio_trials <- plot(h_learn_trials, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)

sd_ratio_trials[[1]] +
  geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
```



## Conclusions

-	Gamified feedback had no effect on response accuracy during practice. Response accuracy was slightly higher in the Control condition for participants in the Points group than for participants in the Progress bar group, possibly due to group differences in spite of random assignment.
-	Correct responses were slower during practice with gamified feedback than in the control condition, particularly so in Block 1.
-	There were no overall effects of the gamification manipulation on total score, number of words practiced, or number of trials completed. However, in the gamified conditions, these outcomes were all better in Block 2 than in Block 1.


## Combined plot
```{r}
((p_learn_acc | p_learn_rt) / (p_learn_trials | p_learn_words | p_learn_score)) +
  plot_layout(guides = "collect") +
  plot_annotation(tag_levels = "a")

ggsave(here("output", "practice_performance_all.png"), width = 7.5, height = 5)
```

Streamlined version:
```{r}
(p_learn_acc | p_learn_rt | p_learn_score) +
  plot_layout(guides = "collect") +
  plot_annotation(tag_levels = "a")

ggsave(here("output", "practice_performance.png"), width = 7.5, height = 3)
```
# Save hypothesis testing output for visualisation
```{r}
fwrite(h_learn_acc$hypothesis, here("output", "hypothesis_tests", "h_learn_acc.csv"))
fwrite(h_learn_rt$hypothesis, here("output", "hypothesis_tests", "h_learn_rt.csv"))
fwrite(h_learn_score$hypothesis, here("output", "hypothesis_tests", "h_learn_score.csv"))
fwrite(h_learn_words$hypothesis, here("output", "hypothesis_tests", "h_learn_words.csv"))
fwrite(h_learn_trials$hypothesis, here("output", "hypothesis_tests", "h_learn_trials.csv"))
```

# Session info
```{r}
sessionInfo()
```
